Electro-osmosis on Anisotropic Super-hydrophobic Surfaces

week ending 26 AUGUST 2011

PHYSICAL REVIEW LETTERS

PRL 107, 098301 (2011)

Electro-osmosis on Anisotropic Superhydrophobic Surfaces Aleksey V. Belyaev1,2 and Olga I. Vinogradova1,2,3 1

2

Department of Physics, M. V. Lomonosov Moscow State University, 119991 Moscow, Russia A. N. Frumkin Institute of Physical Chemistry and Electrochemistry, Russian Academy of Sciences, 31 Leninsky Prospect, 119991 Moscow, Russia 3 ITMC and DWI, RWTH Aachen University, Pauwelsstrasse 8, 52056 Aachen, Germany (Received 15 March 2011; published 23 August 2011)

We give a general theoretical description of electro-osmotic flow at striped superhydrophobic surfaces in a thin double layer limit, and derive a relation between the electro-osmotic mobility and hydrodynamic slip-length tensors. Our analysis demonstrates that electro-osmotic flow shows a very rich behavior controlled by slip length and charge at the gas sectors. In the case of an uncharged liquid-gas interface, the flow is the same or inhibited relative to the flow in a homogeneous channel with a zero interfacial slip. By contrast, it can be amplified by several orders of magnitude provided slip regions are uniformly charged. When gas and solid regions are oppositely charged, we predict a flow reversal, which suggests the possibility of a huge electro-osmotic slip even for electroneutral surfaces. On the basis of these observations we suggest strategies for practical microfluidic devices. DOI: 10.1103/PhysRevLett.107.098301

PACS numbers: 47.57.jd, 68.08.p, 83.50.Lh

Introduction.—Electro-osmotic (EO) ‘‘plug’’ flows are established when an electric field forces the diffuse ionic cloud adjacent to a charged surface in an electrolyte solution into motion. This classical subject of colloid science [1] is currently experiencing a renaissance in micro- and nanofluidics [2,3], which raises the fundamental question of how to pump and mix fluids at micron scales, where pressure-driven flows and inertial instabilities are suppressed by viscosity. Electro-osmosis offers unique advantages in this area of research and technologies, such as low hydrodynamic dispersion, no moving parts, electrical actuation and sensing, energy conversion and storage, and easy integration with microelectronics. Until recently, almost all studies of EO flow have assumed uniform surface charge and no-slip hydrodynamic boundary conditions at the surface. In such a situation the scalar electro-osmotic mobility M1 , which relates an apparent EO ‘‘slip’’ velocity U 1 (outside of the thin double layer) to the tangential electric field Et is given by the classical Smoluchowski formula [4] M1 ¼

U1 q ¼ 1; Et

(1)

where is the viscosity of the solution, q1 is the constant charge density of the no-slip surface, which can be related to the zeta potential across the diffuse (flowing) part of the double layer, 1 ¼ q1 =", where " is the permittivity of the solution, and ¼ 1 D is the inverse Debye screening length, that characterizes the thickness of the electrical double layer (EDL). Recent studies demonstrated the existence of a hydrodynamic slip at a hydrophobic smooth and homogeneous surface, which can be quantified by the slip length b (the distance within the solid at which the flow profile 0031-9007=11=107(9)=098301(4)

extrapolates to zero) [5,6]. The combination of the strategies of EO and hydrophobic slip, can yield enhanced EO flow. For a charge density q2 of the slipping interface, simple arguments show that the EO mobility is given by [7,8]: M2 ¼

U2 q ¼ 2 ð1 þ bÞ: Et

(2)

Since the EO flow amplification scales as (1 þ b), and b can be of the order of tens of nanometers [9–12], for typically nanometric Debye length an order of magnitude enhancement might be expected. It is now natural to assume that a massive amplification of EO flow can be reached on superhydrophobic (SH) surfaces where the effective, in general case tensorial, slip length, beff , could be of the order of several microns [13–15]. The controlled generation of such flows is by no means obvious, since both the slip length and the electric charge distribution on a SH surface are inhomogeneous and often anisotropic. Despite its fundamental and practical significance, EO flow over SH surfaces has received little attention. Recently, [16] investigated EO flow past inhomogeneously charged, flat SH surfaces in the case of thick channels (h L), thin EDL (L 1), and predicted q M ¼ M1 I þ 2 beff (3) q1 by using the Lorentz reciprocal theorem for the Stokes flow and by assuming a perfect slip (b ¼ 1) at gas sectors. Here I is the unit tensor, and we keep the notations q1 and q2 to characterize the surface charge density at the no-slip and slip regions, as above. This expression indicates negligible flow enhancement in case of an uncharged

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Ó 2011 American Physical Society

PRL 107, 098301 (2011)

liquid-gas interface (which has been confirmed by later studies [17,18]), and shows that surface anisotropy generally leads to a tensorial EO response. In this Letter, a general situation of EO flow past SH surfaces with patterns of arbitrary partial slip, is considered (Fig. 1). Our focus is on the canonical EO geometry of a thick parallel-plate channel with a two-component (no-slip and slip) coarse texture, varying on scales larger than the EDL thickness. General theory.—To highlight the effect of anisotropy, we focus on an idealized, flat, periodic, charged, striped SH surface in the Cassie state, sketched in Fig. 1, where the liquid-solid interface has no slip (b1 ¼ 0) and the liquidgas interface has a partial slip (b2 ¼ b, 0 b 1). As a simple estimate, lubricating gas sectors of height e with viscosity g much smaller than [19] have a local slip length b2 eð=g Þ 50e, which can reach tens of m. Let then 1 and 2 ¼ =L be the area fractions of the solid and gas phases with 1 þ 2 ¼ 1. Pressure-driven flow past such stripes has been shown to depend on the direction of the flow, and the eigenvalues of the slip-length tensor [20] read [21] 2 ln½secð L 2 Þ ; 2 L 2 1 þ b ln½secð 2 Þ þ tanð 2 Þ

(4)

2 ln½secð L 2 Þ : L 2 2 1 þ 2b ln½secð 2 2 Þ þ tanð 2 Þ

(5)

bkeff ’

b? eff ’



The EO mobility is represented by 2 2 matrices diagonalized by a rotation [18]. By symmetry, the eigendirections of M correspond to longitudinal ( ¼ 0) and transverse ( ¼ =2) alignment with the applied electric field (Fig. 1), so we need only to compute the eigenvalues, Mk and M? , for these cases. We consider a semi-infinite electrolyte in the region y > 0 above a patterned surface at y ¼ 0 subject to an ^ in the x direction. For nanoscale electric field, Et ¼ Et x, patterns (L < 1 m), we can neglect convection (Pe 1 for a typical ionic diffusivity D), so that c ðx; y; zÞ is independent of the fluid flow [23]. We also assume a weak field (jEt jL j c j) and a weakly charged surface (j c j kB T=ðzeÞ ¼ 25=z mV) for a z:z electrolyte, so that c satisfies the Debye-Hu¨ckel equation with a boundary condition of prescribed surface charge, r2 c ¼ 2 c ;

"@y c ¼ qðx; 0; zÞ:

The fluid flow satisfies Stokes’ equations with an electrostatic body force ^ r2 u ¼ rp þ "2 c Et x;

that coincides with results obtained for the perfect slip (b ¼ 1) stripes [22].

r u ¼ 0;

(8)

with the boundary conditions at y ¼ 0 u t ¼ bðx; zÞ@y ut ;

These expressions depend strongly on a texture period L. When b=L 1 they predict the area-averaged isotropic slip length, b?;k eff ’ 2 b. When b=L 1, expressions (4) and (5) take the form L 2 ? ln sec ; bkeff ’ 2b? beff ’ (6) eff ; 2 2

(7)

y^ u ¼ 0;

(9)

where ut ¼ ux^ þ wz^ is the lateral, and v ¼ y^ u is normal to the surface velocities. We also neglect surface conduction (which tends to reduce EO flow) compared to bulk conduction (Du 1) [23]. Far from the surface, y ! 1, u approaches the EO slip velocity E ¼ M Et and

c ! 0;

@y u ! 0:

(10)

For a longitudinal configuration only the velocity component parallel to Et ¼ Et x^ remains. In case of transverse to applied field stripes, normal velocity v y^ does not vanish due to mass conservation condition in (8), which can significantly modify the EO flow. Rigorous calculations [23] allow one to find exact solutions for Uk;? , and thus obtain the eigenvalues of the EO mobility tensor: M

k;?

¼

M1 bk;? eff

q2 q1 þ q2 b þ1 ; q1 b

(11)

where the effective slip lengths are given by Eqs. (4) and (5). The flow is thus anisotropic and there is a simple relationship between the EO mobility and hydrodynamic slip-length tensors [24]

FIG. 1 (color online). (a) Sketch of the superhydrophobic effective slippage effect on the EO flow. The real situation is approximated by a periodic cell of size L, with patterns of charges and flow boundary conditions. (b) Illustration of tensorial EO response: ¼ =2 corresponds to transverse stripes, whereas ¼ 0 to longitudinal stripes.

beff q2 M ¼ M1 I þ ð1 þ bÞ 1 : b q1

(12)

In the limit of b=L 1 the general expression transforms to Eq. (3). When b=L 1 we get isotropic EO flow

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M ¼ 1 M1 þ 2 M2 :

(13)

Discussion.—To demonstrate examples of very rich and unusual fluid behavior at the SH surface it is instructive to consider some limiting cases of Eq. (12) with different values of q1 and q2 . The results (shown in Fig. 2) are somewhat remarkable. We see, in particular, that if the gas area is uncharged, the EO flow related then only to the charge q1 on the solid-liquid interface is generally inhibited as compared with a homogeneous, solid no-slip surface with uniform charge density [see Fig. 2(a)] b M ¼ M1 I eff : b

(14)

M/M1 (a)

(b)

b/L

M ¼ M1 ½I þ beff ;

which for b=L 1 simply gives M ¼ M1 ½1 2 ð1þbÞ. The calculation results for this situation are presented in Fig. 2(c), and suggest a very rich fluid behavior. We see, in particular, that inhomogeneous surface charge can induce EO flow along or opposite to the field, depending on the fraction of the gas area as shown in Fig. 2(d). Already a very small fraction of the gas sectors would be enough to reverse the effective EO flow. Another striking result is that an electro-neutral surface (1 q1 þ 2 q2 ¼ 0) can generate an extremely large EO slip. With our numerical example this corresponds to 2 ¼ 0:5. In other words, a superhydrophobic surface of average positive charge or even zero charge can induce an EO flow (different for longitudinal and transverse to applied field stripes) in the direction of the applied field as if it is uniformly and

(a)

(d)

(b)

|Uz/Ux|

(c)

φ2

FIG. 2. Eigenvalues of normalized EO mobility. Solid curves represent longitudinal, and dashed—transverse alignment of stripes with electric field. Mk;? =M1 vs amplitude of the local slip b=L for (a) uncharged slip areas (q2 ¼ 0, 2 ¼ 0:5, L ¼ 102 ), (b) uniform charge distribution (q2 ¼ q1 ; 2 ¼ 0:45, L ¼ 103 ) and (c) oppositely charged slip and no-slip areas (q2 ¼ q1 . 2 ¼ 0:35, L ¼ 102 ). Mk;? =M1 plotted against (d) the fraction of gas sectors (b=L ¼ 0:1, q2 ¼ q1 , L ¼ 102 ).

(15)

which might be seen as a natural tensorial analog of Eq. (2). Figure 2(b) includes theoretical results calculated with Eq. (15) for a geometry of stripes, and is intended to demonstrate that the flow is truly anisotropic and can exhibit a large enhancement from effective hydrodynamic slip, possibly by several orders of magnitude. We stress that such an enhancement is possible even at a relatively low gas fraction, i.e., when beff is relatively small [but the amplification factor, (I þ beff ), might be huge]. Also note that in this situation Mk M? ; i.e., the fastest (slowest) direction can correspond only to longitudinal (transverse) stripes. An interesting scenario is expected for oppositely charged solid and gas sectors. If q1 ¼ q2 , then Eq. (12) transforms to b M ¼ M1 I 2 eff beff ; (16) b

M/M1

M/M1

b/L

The situation is very different if the slipping interface carries some net charge, which is not an unreasonable assumption [27]. To gain some insight into the possible EO flow enhancement, we consider first the case of uniform surface charge q1 ¼ q2 , where Eq. (12) gives

|Uz/Ux|

M/M1

We remark and stress that in contrast to common expectations, the situation described by Eq. (14) corresponds to Mk M? ; i.e., the maximal directional mobility is attained in a transverse, and minimal—in the longitudinal direction [26]. When b=L 1 we simply get M ¼ 1 M1 . In other words, the (isotropic) EO mobility shows no manifestation of the slip, being equal to the surface averaged velocity generated by no-slip regions. This result coincides with what is expected for hydrophilic uncharged sectors. In the limit of b=L 1 this inhibition becomes negligibly small, and we obtain the simple results of [16,18], where EO mobility becomes equal to M1 regardless of the orientation or area fraction of the slipping stripes. These results suggest that although the absence of the screening cloud near the gas region tends to inhibit the effective EO slip, the hydrodynamic slip acts to suppress this inhibition.

b/L



PRL 107, 098301 (2011)

/4

/2

b/L

FIG. 3. The ratio of EO velocity components, jUz =Ux j, (at 2 ¼ 0:5, L ¼ 102 ) as a function of (a) angle at b=L ¼ 102 and (b) local slip length at the optimal angle . From top to bottom q2 =q1 ¼ 2, 1, 0.5.

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PRL 107, 098301 (2011)

y/L

1

0.5

0.5

0 -0.5

-0.25

0

x/L

0.25

0.5 -0.5

-0.25

0

x/L

0.25

0.5 -0.5

-0.25

0

0.25

0.5

x/L

FIG. 4 (color online). Streamlines of the EO flow computed at 2 ¼ 0:35 and L ¼ 102 for q2 =q1 ¼ 0:43 and ¼ =2. The origin of coordinates coincides with the center of the gas region. From left to right the local slip length is b=L ¼ 0:1, 0.5, and 102 .

negatively charged. These findings are similar to those of [4,28] that the electrokinetic mobility depends on the charge distribution on the object, and not solely on its total charge. However, in our case the flow is dramatically amplified due to hydrodynamic effective slip. These results may guide the design of SH surfaces for transverse electrokinetic flows in microfluidic devices [28]. As we have shown above, effective EO mobility of anisotropic striped surfaces is generally tensorial, due to secondary flow transverse to the direction of the applied electric field. Anisotropy (jUz =Ux j) is maximized in the direction of max [as it is seen in Fig. 3(a)] and requires that q2 =q1 and b are as large as possible. In a thick SH channel a transverse plug EO flow seems to be a very fruitful direction compared to transverse hydrodynamic phenomena, where flow is ‘‘twisted’’ only near the wall [29]. Another mixing mechanism is related to the formation of patterns of steady convective rolls on the scale proportional to the texture period (Fig. 4). This can happen in the situation of oppositely charged and transverse to applied field stripes. Figure 4 illustrates the effect of the local slip on the morphology of the steady rolls formation. We see, in particular, that increase of b leads first to the appearance of additional convective patterns near no-slip areas, and then to a transition to a flow morphology, where recirculation of a fluid is observed only at the no-slip regions. This, in turn, induces the flow reversal [as in Fig. 2(d)]. Concluding remarks.—We have described EO flow on inhomogeneously charged and slipping anisotropic surfaces. Our analysis provided the necessary tools to describe a significant modification of EO phenomena on SH surfaces: to quantify the inhibition and enhancement of flow, the transition from its anisotropy to isotropy, onsets of convective rolls formation and a relevant flow reversal, which can generate a huge EO slip even in the situation of a zero mean charge. Our results may find numerous applications in microfluidic lab-on-a-chip devices. This research was supported by the RAS through its priority program ‘‘Assembly and Investigation of Macromolecular Structures of New Generations.’’


[1] J. Lyklema, Fundamentals of Interface and Colloid Science. Volume II: Solid-Liquid Interfaces (Academic Press Limited, San Diego, CA, 1995). [2] H. A. Stone, A. D. Stroock, and A. Ajdari, Annu. Rev. Fluid Mech. 36, 381 (2004). [3] J. C. T. Eijkel and A. van den Berg, Microfluid Nanofluid 1, 249 (2005). [4] J. L. Anderson, Annu. Rev. Fluid Mech. 21, 61 (1989). [5] O. I. Vinogradova, Int. J. Miner. Process. 56, 31 (1999). [6] E. Lauga, M. P. Brenner, and H. A. Stone, in Handbook of Experimental Fluid Dynamics edited by C. Tropea, A. Yarin, and J. F. Foss (Springer, NY, 2007), Chap. 19, pp. 1219–1240. [7] V. M. Muller, I. P. Sergeeva, V. D. Sobolev and N. V. Churaev, Colloid J. USSR 48, 606 (1986). [8] L. Joly, C. Ybert, E. Trizac, and L. Bocquet, Phys. Rev. Lett. 93, 257805 (2004). [9] O. I. Vinogradova and G. E. Yakubov, Langmuir 19, 1227 (2003). [10] C. Cottin-Bizonne, B. Cross, A. Steinberger, and E. Charlaix, Phys. Rev. Lett. 94, 056102 (2005). [11] L. Joly, C. Ybert, and L. Bocquet, Phys. Rev. Lett. 96, 046101 (2006). [12] O. I. Vinogradova, K. Koynov, A. Best, and F. Feuillebois, Phys. Rev. Lett. 102, 118302 (2009). [13] J. Ou and J. P. Rothstein, Phys. Fluids 17, 103606 (2005). [14] P. Joseph, C. Cottin-Bizonne, J. M. Benoıˇ, C. Ybert, C. Journet, P. Tabeling, and L. Bocquet, Phys. Rev. Lett. 97, 156104 (2006). [15] P. Tsai, A. M. Peters, C. Pirat, M. Wessling, R. G. H. Lammerting, and D. Lohse, Phys. Fluids 21, 112002 (2009). [16] T. M. Squires, Phys. Fluids 20, 092105 (2008). [17] D. M. Huang, C. Cottin-Bizzone, C. Ybert, and L. Bocquet, Phys. Rev. Lett. 101, 064503 (2008). [18] S. S. Bahga, O. I. Vinogradova, and M. Z. Bazant, J. Fluid Mech. 644, 245 (2010). [19] O. I. Vinogradova, J. Colloid Interface Sci. 169, 306 (1995). [20] M. Z. Bazant and O. I. Vinogradova, J. Fluid Mech. 613, 125–134 (2008). [21] A. V. Belyaev and O. I. Vinogradova, J. Fluid Mech. 652, 489 (2010). [22] E. Lauga and H. A. Stone, J. Fluid Mech. 489, 55 (2003). [23] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.107.098301 for details of the analysis. [24] Note some similarity with [25], where a thermophoretic mobility was found to depend on b and b=R. [25] J. Morthomas and A. Wu¨rger, J. Phys. Condens. Matter 21, 035103 (2009). [26] This surprising observation likely reflects the fact, that the reduction of the interfacial stress (and the mobility) due to slip in the absence of the diffuse layer should be smaller for transverse stripes since it is partly compensated by the local pressure drop. [27] V. Tandon, S. K. Bhagavatula, W. C. Nelson, and B. J. Kirby, Electrophoresis 29, 1092 (2008). [28] A. Ajdari, Phys. Rev. E 65, 016301 (2001). [29] O. I. Vinogradova and A. V. Belyaev, J. Phys. Condens. Matter 23, 184104 (2011).

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